kaluza-klein and coset spaces
TRANSCRIPT
-
8/11/2019 Kaluza-Klein and Coset Spaces
1/48
Kaluza-Klein and Coset Spaces
Alexander Bols
June 17, 2014
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 1 / 16
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
2/48
Overview
Compactification on Tori.
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 2 / 16
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
3/48
Overview
Compactification on Tori.
T2 reduction of supergravity : the resulting scalars and theirsymmetries
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 2 / 16
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
4/48
Overview
Compactification on Tori.
T2 reduction of supergravity : the resulting scalars and theirsymmetries
Extension to Tn.
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 2 / 16
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
5/48
Overview
Compactification on Tori.
T2 reduction of supergravity : the resulting scalars and theirsymmetries
Extension to Tn.
Summary of symmetry groups and their isotropy groups.
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 2 / 16
http://find/http://goback/ -
8/11/2019 Kaluza-Klein and Coset Spaces
6/48
Compactification on tori
Consider gravity in Ddimensions. The metric has D2 components
gij=gij(x1, , xD). (1)
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 3 / 16
http://find/http://goback/ -
8/11/2019 Kaluza-Klein and Coset Spaces
7/48
Compactification on tori
Consider gravity in Ddimensions. The metric has D2 components
gij=gij(x1, , xD). (1)
We can obtain a field theory on a D 1 dimensional space by
compactifying one coordinate on T1.
xD xD+ 2LZ (2)
with L the radius of the torus.
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 3 / 16
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
8/48
Compactification on tori
Consider gravity in Ddimensions. The metric has D2 components
gij=gij(x1, , xD). (1)
We can obtain a field theory on a D 1 dimensional space by
compactifying one coordinate on T1.
xD xD+ 2LZ (2)
with L the radius of the torus.
Now the fields can be written as a Fourier series
gij(x1, , xD) =n
g(n)ij (x1, , xD1)e
inxD/L (3)
with L the circumference of the torus.Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 3 / 16
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
9/48
-
8/11/2019 Kaluza-Klein and Coset Spaces
10/48
Compactification on tori
The terms with n= 0 correspond to massive fields with
M n/L. (4)
We assume L to be tiny, so the n = 0 modes can be ignored.
We end up with D2 fields
g(0)ij =g
(0)ij (x1, , xD1) (5)
in D 1 dimensions.
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 4 / 16
http://find/http://goback/ -
8/11/2019 Kaluza-Klein and Coset Spaces
11/48
Compactification on tori
The terms with n= 0 correspond to massive fields with
M n/L. (4)
We assume L to be tiny, so the n = 0 modes can be ignored.
We end up with D2 fields
g(0)ij =g
(0)ij (x1, , xD1) (5)
in D 1 dimensions.
The fields g
(0)
i,D form a vector field.
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 4 / 16
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
12/48
Compactification on tori
The terms with n= 0 correspond to massive fields with
M n/L. (4)
We assume L to be tiny, so the n = 0 modes can be ignored.
We end up with D2 fields
g(0)ij =g
(0)ij (x1, , xD1) (5)
in D 1 dimensions.
The fields g
(0)
i,D form a vector field.The field g
(0)D,D is a scalar field.
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 4 / 16
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
13/48
Compactification on tori
The terms with n= 0 correspond to massive fields with
M n/L. (4)
We assume L to be tiny, so the n = 0 modes can be ignored.
We end up with D2 fields
g(0)ij =g
(0)ij (x1, , xD1) (5)
in D 1 dimensions.
The fields g
(0)
i,D form a vector field.The field g
(0)D,D is a scalar field.
The remaining (D 1)2 components form a metric tensor for theD 1 dimensional space.
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 4 / 16
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
14/48
T2 scalar sector
The same procedure can be repeated to obtain ever lower dimensionaltheories of gravity couples to more and more matter fields.
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 5 / 16
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
15/48
T2 scalar sector
The same procedure can be repeated to obtain ever lower dimensionaltheories of gravity couples to more and more matter fields.
We look at supergravity in 11 dimensions.
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 5 / 16
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
16/48
T2 scalar sector
The same procedure can be repeated to obtain ever lower dimensionaltheories of gravity couples to more and more matter fields.
We look at supergravity in 11 dimensions.
Some of those matter fields are scalars. The scalar sector resulting
from compactification on T2 is
L = 1
2()2
1
2()2
1
2e2()2. (6)
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 5 / 16
http://goforward/http://find/http://goback/ -
8/11/2019 Kaluza-Klein and Coset Spaces
17/48
T2 scalar sector
The same procedure can be repeated to obtain ever lower dimensionaltheories of gravity couples to more and more matter fields.
We look at supergravity in 11 dimensions.
Some of those matter fields are scalars. The scalar sector resulting
from compactification on T2 is
L = 1
2()2
1
2()2
1
2e2()2. (6)
The field only appears in the first term. It is decoupled from theothers and has a shift symmetry R.
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 5 / 16
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
18/48
SL(2,R) symmetry
The other two fields are described by
L = 1
2()2
1
2e2()2 =
2()2 (7)
where =+ie
.
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 6 / 16
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
19/48
SL(2,R) symmetry
The other two fields are described by
L = 1
2()2
1
2e2()2 =
2()2 (7)
where =+ie
.It can be checked that this Lagrangian is invariant under
a+b
c+d (8)
with ad bc= 1.
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 6 / 16
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
20/48
SL(2,R) symmetry
The other two fields are described by
L = 1
2()2
1
2e2()2 =
2()2 (7)
where =+ie
.It can be checked that this Lagrangian is invariant under
a+b
c+d (8)
with ad bc= 1.
These transformations from a group isomorphic to SL(2,R).
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 6 / 16
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
21/48
Lagrangian emerges from the symmetry group
The generators of sl(2,R) have 2 2 representation
H=
1 00 1
; E+ =
0 10 0
; E=
0 01 0
. (9)
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 7 / 16
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
22/48
Lagrangian emerges from the symmetry group
The generators of sl(2,R) have 2 2 representation
H=
1 00 1
; E+ =
0 10 0
; E=
0 01 0
. (9)
Consider the matrix
V=eHeE+ =e/2 e/2
0 e/2. (10)
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 7 / 16
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
23/48
Lagrangian emerges from the symmetry group
The generators of sl(2,R) have 2 2 representation
H=
1 00 1
; E+ =
0 10 0
; E=
0 01 0
. (9)
Consider the matrix
V=eHeE+ =e/2 e/2
0 e/2. (10)
The Lagrangian can then be written as
L =14
Tr M1M = 12
()2 12e2()2 (11)
wereM = VTV. (12)
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 7 / 16
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
24/48
SL(2,R) symmetry is manifest
This Lagrangian is manifestly invariant underV V = V (13)
with SL(2,R).
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 8 / 16
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
25/48
SL(2,R) symmetry is manifest
This Lagrangian is manifestly invariant under
V V = V (13)
with SL(2,R).
The new matrix V
should encode transformed scalar fields. To seewhat the new scalars are, we put V back in upper triangular form bymultiplying with a uniqueorthogonal matrix
V = OV = OV. (14)
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 8 / 16
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
26/48
SL(2,R) symmetry is manifest
This Lagrangian is manifestly invariant under
V V = V (13)
with SL(2,R).
The new matrix V
should encode transformed scalar fields. To seewhat the new scalars are, we put V back in upper triangular form bymultiplying with a uniqueorthogonal matrix
V = OV = OV. (14)
Again, the Lagrangian is manifestly invariant under thistransformation.
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 8 / 16
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
27/48
Scalar moduli space is coset
The group SL(2,R) represented by acts transitively on the scalarmanifold at a fixed point in spacetime
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 9 / 16
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
28/48
Scalar moduli space is coset
The group SL(2,R) represented by acts transitively on the scalarmanifold at a fixed point in spacetime
But there are many that lead to the same transformed fields. Wehave to compensate with a Orthogonal matrix to get a uniquetransformation connecting any two points on the scalar manifold.
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 9 / 16
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
29/48
Scalar moduli space is coset
The group SL(2,R) represented by acts transitively on the scalarmanifold at a fixed point in spacetime
But there are many that lead to the same transformed fields. Wehave to compensate with a Orthogonal matrix to get a uniquetransformation connecting any two points on the scalar manifold.
Therefore, each point on the scalar manifold can be identified withthe unique transformation that maps this point to the origin.
scalar manifold SL(2,R)/O(2). (15)
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 9 / 16
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
30/48
Scalar moduli space is coset
The group SL(2,R) represented by acts transitively on the scalarmanifold at a fixed point in spacetime
But there are many that lead to the same transformed fields. Wehave to compensate with a Orthogonal matrix to get a uniquetransformation connecting any two points on the scalar manifold.
Therefore, each point on the scalar manifold can be identified withthe unique transformation that maps this point to the origin.
scalar manifold SL(2,R)/O(2). (15)
Note that O(2) is the maximally compact subgroup ofSL(2,R)
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 9 / 16
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
31/48
Generalization to Tn
For compactification on Tn
we get a scalar sector of the form
L = 1
2d
1
2
i
-
8/11/2019 Kaluza-Klein and Coset Spaces
32/48
Generalization to Tn
For compactification on Tn
we get a scalar sector of the form
L = 1
2d
1
2
i
-
8/11/2019 Kaluza-Klein and Coset Spaces
33/48
Generalization to Tn
For compactification on Tn
we get a scalar sector of the form
L = 1
2d
1
2
i
-
8/11/2019 Kaluza-Klein and Coset Spaces
34/48
Dilaton vectors are positive roots
In the Iwasawa decomposition, gHgNgeneralizes our V and gKgeneralizes the orthogonal transformation.
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 11 / 16
Dil i i
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
35/48
Dilaton vectors are positive roots
In the Iwasawa decomposition, gHgNgeneralizes our V and gKgeneralizes the orthogonal transformation.
Vis a representative of the coset G/K
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 11 / 16
Dil i i
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
36/48
Dilaton vectors are positive roots
In the Iwasawa decomposition, gHgNgeneralizes our V and gKgeneralizes the orthogonal transformation.
Vis a representative of the coset G/K
As in the case ofT2, we find that the scalar Lagrangian can be
written in terms ofV
L =1
4Tr (V#V)1(V#V). (17)
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 11 / 16
Dil t t iti t
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
37/48
Dilaton vectors are positive roots
In the Iwasawa decomposition, gHgNgeneralizes our V and gKgeneralizes the orthogonal transformation.
Vis a representative of the coset G/K
As in the case ofT2, we find that the scalar Lagrangian can be
written in terms ofV
L =1
4Tr (V#V)1(V#V). (17)
The dilaton vectors bij and aijkarise from the commutators ofelements the Cartan subalgebra and positive root generators. Theyare precisely the positive roots ofG!
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 11 / 16
F L i t t
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
38/48
From Lagrangian to symmetry group
We now proceed as follows
Find a set of dilaton vectors that can serve as simple roots ofG.
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 12 / 16
F L g gi t s t g
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
39/48
From Lagrangian to symmetry group
We now proceed as follows
Find a set of dilaton vectors that can serve as simple roots ofG.
Construct the corresponding Dynkin diagram.
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 12 / 16
From Lagrangian to symmetry group
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
40/48
From Lagrangian to symmetry group
We now proceed as follows
Find a set of dilaton vectors that can serve as simple roots ofG.
Construct the corresponding Dynkin diagram.Gis the normal real form corresponding to the diagram.
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 12 / 16
From Lagrangian to symmetry group
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
41/48
From Lagrangian to symmetry group
We now proceed as follows
Find a set of dilaton vectors that can serve as simple roots ofG.
Construct the corresponding Dynkin diagram.Gis the normal real form corresponding to the diagram.
Kis the maximally compact subgroup ofG.
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 12 / 16
The simple roots
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
42/48
The simple roots
The dilaton vectors have a very simple structure.
The simple roots are bi,i+1 and a123.
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 13 / 16
The simple roots
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
43/48
The simple roots
The dilaton vectors have a very simple structure.
The simple roots are bi,i+1 and a123.
All simple roots have length 2.
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 13 / 16
The simple roots
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
44/48
The simple roots
The dilaton vectors have a very simple structure.
The simple roots are bi,i+1 and a123.All simple roots have length 2.
The bij form a chain
bi,i+1 bi+1,i+2 = 2 (18)
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 13 / 16
The simple roots
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
45/48
The simple roots
The dilaton vectors have a very simple structure.
The simple roots are bi,i+1 and a123.All simple roots have length 2.
The bij form a chain
bi,i+1 bi+1,i+2 = 2 (18)
and a123 connects to b34 only
a123 bi,i+1 = 2i,3 (19)
i.e. for n= 3, the root a123 is disconnected from the diagram,otherwise it connects to b34.
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 13 / 16
The Dynkin diagrams
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
46/48
The Dynkin diagramsn Dynkin diagram algebra
2
b12
E1 =A1 =sl
(2,R
)3
b12 b23 a123 E3 =A1 A2 = sl(2,R) sl(3,R)
4
b12 b23
b34 a123 E4 =A4 = sl(5,R)
5
b12 b23
b34 b45
a123
E5=D5
6
b12 b23
b34 b45
b56
a123
E6
7
b12 b23
b34 b45
b56 b67
a123
E7
8
b12 b23
b34 b45
b56 b67
b78
a123
E8
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 14 / 16
The corresponding groups
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
47/48
The corresponding groups
(see course notes, Table 3)
n algebra G K
2 A1 GL(2) O(2)3 A1 A2 SL(3) SL(2) SO(3) SO(2)
4 A4 SL(5) SO(5)5 D5 O(5, 5) O(5) O(5)6 E6 E6(+6) USp(8)7 E7 E7(+7) SU(8)
8 E8 E8(+8) SO(16)
These are the global symmetry groups (G) and their isotropy groups (K)of the scalar sectors resulting from compactification on tori Tn.
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 15 / 16
Questions?
http://find/ -
8/11/2019 Kaluza-Klein and Coset Spaces
48/48
Questions?
Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 16 / 16
http://find/